Abstract
Without any smallness assumption, we prove the global unique solvability of the 2-D incompressible inhomogeneous Navier-Stokes equations with initial data in the critical Besov space, which is almost the energy space in the sense that they have the same scaling in terms of this 2-D system.
Similar content being viewed by others
References
Abidi, H.: Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique. Rev. Mat. Iberoam. 23(2), 537–586, 2007
Abidi, H., Gui, G., Zhang, P.: On the wellposedness of \(3-\)D inhomogeneous Navier-Stokes equations in the critical spaces. Arch. Rational Mech. Anal. 204, 189–230, 2012
Abidi, H., Paicu, M.: Existence globale pour un fluide inhomogéne. Ann. Inst. Fourier (Grenoble) 57, 883–917, 2007
Abidi, H., Zhang, P.: On the global well-posedness of \(2\)-D inhomogeneous incompressible Navier-Stokes system with variable viscous coefficient. J. Differ. Equ. 259, 3755–3802, 2015
Kazhikov, A.V.: Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, (Russian) Dokl. Akad. Nauk SSSR 216, 1008–1010, 1974
Bahouri, H., Chemin, J.Y.; Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Grundlehren der Mathematischen Wissenschaften, 2010
Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Cham, 1976
Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246, 1981
Chemin, J.-Y.: Théorémes d’unicité pour le systéme de Navier-Stokes tridimensionnel. J. Anal. Math. 77, 27–50, 1999
Chemin, J.-Y., Lerner, N.: Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes. J. Differ. Equ. 121, 314–328, 1995
Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm. Partial Differ. Equ. 26, 1183–1233, 2001
Danchin, R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. R. Soc. Edin. Sect. A 133, 1311–1334, 2003
Danchin, R.: Local and global well-posedness resultats for flows of inhomogenenous viscous fluids. Adv. Differ. Equ. 9, 353–386, 2004
Danchin, R.: The inviscid limit for density-dependent incompressible fluids. Anneles de la Faculté de Sciences des Toulouse Sér. 15, 637–688, 2006
Danchin, R., Mucha, P.B.: The incompressible Navier-Stokes equations in vacuum. Comm. Pure. Appl. Math. 72(7), 1351–1385, 2019
Danchin, R., Mucha, P.B.: A Lagrangian approach for the incompressible Navier-Stokes equations with variable density. Comm. Pure. Appl. Math. 65, 1458–1480, 2012
DiPerna, R. J., Lions, P. L.: Equations différentielles ordinaires et équations de transport avec des coefficients irréguliers. In Séminaire EDP 1988-1989, Ecole Polytechnique, Palaiseau, 1989
Fleet, T.M.: Differential analysis. Cambridge University Press, Cambridge, 1980
Haspot, B.: Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity. Annales de l’Institut Fourier 62(5), 1717–1763, 2012
Ladyženskaja, O.A., Solonnikov, V.A.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. (Russian) Boundary value problems of mathematical physics, and related questions of the theory of functions. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 52(52–109), 218–219, 1975
Lions, P. L.: Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996
Paicu, M., Zhang, P., Zhang, Z.: Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density. Commun. Partial Differ. Equ. 38(7), 1208–1234, 2013
Peetre, J.: New thoughts on Besov spaces, Duke University Mathematical Series 1, Durham N. C., 1976
Planchon, F.: An extension of the Beale-Kato-Majda criterion for the Euler equations. Comm. Math. Phys. 232, 319–326, 2003
Triebel, H.: Theory of Function Spaces, Monograph in mathematics, vol. 78. Birkhauser Verlag, Basel, 1983
Funding
We thank the anonymous referee for the profitable suggestions. G. Gui is supported in part by the National Natural Science Foundation of China under the Grants 11571279 and 11931013.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P.-L. Lions
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Abidi, H., Gui, G. Global Well-Posedness for the 2-D Inhomogeneous Incompressible Navier-Stokes System with Large Initial Data in Critical Spaces. Arch Rational Mech Anal 242, 1533–1570 (2021). https://doi.org/10.1007/s00205-021-01710-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-021-01710-y